State multipoles

The interpretation of the density (and efficiency) matrices is rather obvious the diagonal elements represent the probabilities of finding a certain magnetic quantum number of the ensemble, and the non-diagonal elements contain the phase information of the system for the corresponding different magnetic quantum numbers. However, in calculations with these matrices one still has to work out the cumbersome summations over these quantum numbers. It is more convenient to replace the density (and efficiency) matrices by statistical and efficiency tensors, also called state multipoles. 

Let us analyze how to find the excited state multipole moments bPq-As explained in the previous paragraph, at excitation by weak light the probability density pb(0, state angular momentum distribution is proportional to the absorption probability G(0,multipole moments, Pq of an excited level b can be found as... 

Table 3.3. Limiting values of the ratio aPo/aPo °f various rank ground state multipole moments aPo and the population aPo for an infinitely large parameter X — 00...

The system of equations obtained, (5.22) and (5.23), in broad line approximation in many cases allows us to carry out the analysis of non-linear optical pumping of both atoms and molecules in an external magnetic field. Some examples will be considered in Section 5.5, among them the comparatively unexplored problem of transition from alignment to orientation under the influence of the dynamic Stark effect. But before that we will return to the weak excitation and present, as examples, some cases of the simultaneous application of density matrix equations (5.7) and expansion over state multipoles (5.20). 

Bain, A.J. and McCaffery, A.J. (1984). Complete determination of the state multipoles of rotationally resolved polarized fluorescence using a single experimental geometry, J. Chem. Phys., 80, 5883-5892. 

Heck and Williams (1987) observed quantum beats in the decay of the n = 2 states of atomic hydrogen. In the presence of an external electric field the 2s and 2p states can be mixed and their correlation measured. With an applied field of 250 Vcm the modulation periods should be 0.1 ns and 0.6 ns. They were able to observe the second beat period corresponding to interference between the states which reduce to 2si/2 and 2pi/2 in the field-free limit. Williams and Heck (1988) were able to use the technique to determine many of the n = 2 state multipoles (section 8.2.4). 

The above description of the excited states in terms of excitation amplitudes is frame and basis set dependent. A more convenient description is in terms of state multipoles. It can be generalised to excited states of different orbital angular momentum and provides more physical insight into the dynamics of the excitation process and the subsequent nature of the excited ensemble. The angular distribution and polarisation of the emitted photons are closely related to the multipole parameters (Blum, 1981). The representation in terms of state multipoles exploits the inherent symmetry of the excited state, leads to simple transformations under coordinate rotations, and allows for easy separation of the dynamical and geometric factors associated with the radiation decay. 

State multipoles are components of a spherical tensor, which is the following tensor product (3.99), averaged over unobserved magnetic quantum numbers. 

Hermiticity imposes further constraints. For / = 1 there are only five independent multipoles, e.g. (T ), (T/), T ), T ), Tq), with (T/) imaginary and the components of the alignment tensor real (Blum, 1981). When reflection invariance holds in the collision plane the state multipoles can be related to the orientation 0 and alignment parameters A, first introduced by Happer (1972) and Fano and Macek (1973), through... 

For completely coherent excitation and / = /= one obtains the following relation between the state multipoles and the parameters a, X, % (8.21). 

The Stokes parameters are directly related to the state multipoles characterising the ensemble of excited-state atoms. Explicit equations are given by Blum and Kleinpoppen (1979), who also give equations where the effects of fine and hyperfine structure are taken into account. As discussed in subsection 8.1.3 five parameters are in general required for determining the excited state. These can be the cross section a and the four Stokes parameters, which are all independent of each other. 

Equations similar to (8.32) can be derived in terms of state multipoles or other parameters. They completely specify the radiation field and show how the properties of this field are related to the dynamical observables of the collision. In practice the time evolution of the excited state is not measured, since it is of no interest in determining X and x, and the total integrated intensity is usually determined experimentally. This means that the factor in C is replaced by 1/y. 

The measurement of the complete set of state multipoles for the n=2 states of hydrogen at 350 eV has been reported by Williams and Heck (1988). The electron scattering angle was 3°. The scattered electrons with 10.2... 

Table 8.4. State multipoles at Eo=350 eV, 6=3° for the n=2 excitation of hydrogen using positive, negative and zero electric fields. The experimental data are due to Williams and Heck (1988). Errors in the final significant figures are given in parentheses. Calculations are CCO, method of Bray, Madison and McCarthy (1990) and vWW, van Wyngaarden and Walters (1986)...

Table 8.4 shows the state multipoles in comparison with the coupled-channels-optical calculation (Bray, Madison and McCarthy, 1990) and the pseudostate calculation of van Wyngaarden and Walters (1986). 

Due to parity conservation A a)x = 0 for a = 45° and 135°. These parameters can be related to the state multipoles T j)) (8.22) describing the atomic state, which depend on the electron polarisation components as follows (Bartschat et al, 1981)... 

Here we have simplified the state multipole notation (8.22) to show only the angular momentum of the excited state. 

Since the absolute differential cross section for scattering by unpolarised electrons was not determined by Sohn and Hanne, they analysed their results using normalised state multipoles defined by... 

Fig. 9.10 shows the normalised state multipoles derived from the Stokes and asymmetry parameters compared with the results of the R-matrix... 

Fig. 9.10. Normalised state multipoles Tkq plotted against scattering angle for electron-impact excitation of Hg (6 Pi) at 8 eV (from Sohn and Hanne, 1992). Curve relativistic i -matrix calculation (Bartschat, 19911)).

In the case of valence band photoemission, the atomic model cannot directly be applied to the numerical estimation of the effect, but rather for a qualitative consideration only. For valence bands, the initial state is no longer described by a single spinor spherical harmonic as it was done in [32] but it can be expanded for a certain k value in a series of spherical harmonics [44] due to their completeness. This procedure will influence the values of the state multipoles and the dipole matrix elements in Eq. 5.6, but the general Eqs. 5.5 and 5.7 will remain unchanged. In particular, they should correcdy describe the dependence of MDAD on the angle of photon incidence. 

An example of a more recent experiment is the smdy of hyperfine structure of highly excited levels of neutral copper atoms, which yielded the magnetic-dipole and electric-quadmpole interaction constants. The experimental results allowed a comparison with theoretical calculations based on multiconfiguration Hartree-Fock methods [858]. Measurements of lifetimes [859] and of state multipoles using level-crossing techniques can be found in [860]. 

G. von Oppen, measurements of state multipoles using level crossing techniques. Comments At. Mol. Phys. 15, 87 (1984)... 

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